Monday, June 24th 2002. Reading: Cosmic Perspective Chapters S3 & 17
When accelerations are involved the situation takes on another face altogether.
Imagine that You and Jackie are floating around in space and You suddenly fire your engines and produce an acceleration up at 9.8 m/s/s.
You cannot consider yourself to be at rest while Jackie accelerates away from You. You feel weight and can see that Jackie is floating freely. You should also be floating free if You are not in motion and she should feel weight if she were accelerating.
It would certainly seem that there is no relativity here. But Einstein thought that the laws of physics should be the same for all observers no matter if they are feeling forces or not.
In 1907 Einstein came upon a great revelation. Whenever you feel weight (as opposed to weightlessness), you can equaly well attribute it to the effects of either acceleration or gravity. This idea is called the equivalence prinicple: The effects of gravity are exactly equivalent to the effects of acceleration.
Imagine that you are sitting in a closed room with the shades drawn. If your room was moved to outerspace and accelerated at a rate of 9.8 m/s/s you would not notice anything different.
If you did physics experiments by dropping balls and so forth you would yeild the same results as you did when the room was on Earth.
Return to You and Jackie. You think that you must be accelerating upward. But You could equivalently say that your ship is at rest on the Earth's surface, perched on a cliff say, and that is why you feel weight. The reason You see Jackie freely floating is that she is falling toward the Earth. She is in freefall just like space shuttle astronauts.
The physics of the two situations are exactly equivalent. So the equivalence principle allows us to state that all motion is relative. This will lead us to some startling discoveries about the nature of space and time!
Some alternative phrasing of the Equivalence Principle:
Spacetime is a 4-D continuum in which the independent directions of motion are up-down, forward-back, left-right, and through time. Events occur in 3 spatial dimensions and at one point in time. In the whole continuum objects are streched out in time as well. If you were a 4-D being in this continuum you could look through time as easily as you look to your right.
In 3-D we can only see projections of 4-D. And the projections can look very different from different point of view. As an example. Imagine a book. Everyone can come measure it and agree on its dimensions. But it can be viewed in projection into 2-D as very many things.
So the heart of relativity is then that different observers are seeing different projections of 4-D spacetime into 3-D space.
"Space is different for different observers. Time is different for different observers. Spacetime is the same for everyone." - Taylor, Wheeler, Freeman.
We must look at lower dimensional analogies to get a feel for what this means since it is not only impossible to imagine 4-D objects, but 4-D space curved into some 5th dimension (a hyperspace).
Consider good old 2-D geometry.
In a flat space with no curvature (a plane):
But in a space which is positively curved (like a sphere):
And in a space that is negatively curved (like a saddle):
If space can be curved and geometry is different in differently curved spaces what does motion in a straight line mean?
It means taking the shortest path.
In a curved space that is going to be a curved line. In relativity that will be a line along which you feel no net forces. A force-free frame travels on a straight line.
A Freely-Floating frame is a force-free frame, and by the equivalence principle so is a Freely-Falling frame.
So if you are falling toward the Earth you are feeling no net forces as you are in freefall, and thus you are on the shortest path toward the center of the Earth. If you wish to get to the other side of the Earth (say from some high poing way up in orbit) the shortest path would not be through the center of the Earth. (Once you pass the center of the Earth you feel a net force back toward the Earth). The shortest path is along the orbit (which also has no net forces acting on it because gravitaional acceleration and centripetal acceleration balance each other).
If we could draw what this situation looks like we could imagine that the 4-Dimensions of spacetime are compressed into a 2-D rubber sheet. An object is placed within the sheet and warps its shape. The curvature of the sheet would then be into the 3rd dimension. And we can see how the orbit would be a shorter path.
We call this an Embedding Diagram.
The strength of Gravity depends upon the amount of spacetime curvature. The amount of spacetime curvature depends on the amount of mass/energy present within.
Orbits are the result of objects passing through curved spacetime.
We can explain the the descrepency between the amount of precession of Mercury's orbit and the amount predicted by Newton's laws. Newton's laws assume the distance around the Sun at a given radius will behave as in plane geometry and hence overestimate the length of the orbit.
General Relativity predicts the precise amount of precession that is actually observed.
Imagine You and Jackie are in a stretch-limo spaceship. You are in the front and Jackie is in the back.
Each of you has a flashing watch perfectly synchronized. The spaceship begins accelerating.
As the ship accelerates You will be carried away from Jackie's flashes and so they are stretched out in time. Thus You see her time to run slower than your own.
Likewise, Jackie is being carried into Your flashes making your time seem to run faster than her own.
Since acceleration is equivalent to gravity we can imagine the same situation with the long ship on Earth. At higher altitudes time runs faster than at lower ones.
Thus, time runs slower in a stronger gravitational field!
This will mean that light will appear to travel curved paths around massive objects. This is called Gravitational Lensing.
The image of an object will appear deflected by the gravity of the foreground object. This can lead to multiple images and even rings if the geometry of the situation is just right.
This has been observed numerous times and is now used to search for different forms of Dark Matter.
This leads to light's frequency appearing to become lower as it escapes the gravity of an object. Hence the light appears more red and this becomes a 2nd kind of redshift: Gravitational Redshift.
Note that because the light is moving toward lower frequency (and hence longer wavelengths) it is also redshifted to lower energy:
One can alternatively think of gravitational redshift arising from the light losing energy as it climbs out of the gravitational field (similar to the way a rocket loses kinetic energy to potential energy).
Gravitational waves have as yet never been detected, because the effect they cause is very small, and the method of detection very tricky. Detectors are currently being built and hope to see gravitaional waves coming from systems that are predicted to put out tons of waves.
Some systems are binary neutron stars in close orbit. Spacetime is warped severly about a neutron star and in a system such as this strong gravity waves are expected.
The waves carry orbital energy away from the system and would cause the binary orbit to decay to smaller and smaller sizes. By Kepler's 3rd law recall that this means a faster orbital period.
One such system: The Hulse-Taylor binary neutron star system. The decay of the orbit is observed to decrease the period precisely by the amount General Relativity predicts.
If light cannot escape a black hole, then nothing can.
The radius at which light cannot escape from is called the Schwarzchild radius:
This surface at this radius is called the Event Horizon. It is called this because beyond this surface all events must be forever unknown.
Imagine a journey into a black hole as seen by two observers: You and Jackie
If she now heads back toward the Event Horizon with the intention of crossing over, you will never see her cross over. The closer she gets to the Horizon the slower her time seems to run and the more redshifted her light (hence the larger the telescope you will need). At the event horizon time is frozen. Eventually, you will either get bored and leave or die while waiting to see her cross.
She had better watch out, however! Tidal forces (the differential pull of gravity) would be so great over the length of her body that she will be stretched into a string. He feet are feel a much stronger gravity than her head.
The effect is weaker the bigger the black hole is. She would be okay if this were a supermassive black hole like those that live in the center of galaxies.
Once she is in that is it. We can predict based on General Relativity what the interior of a black hole might be like, but she can't come back out to tell us if it is true or not. Maybe there is a singularity at the center: a single point where all of the mass is concentrated. Pure Space may give way to pure time at certain points. There is just no way to know.
This is the idea behind a worm hole. It is a tunnel that connects two vastly different regions of spacetime via a short-cut through hyperspace.
Worm holes are only predicted to be possible. They have never been observed and some suspect that there may be no natural way for them to form. They might however be possible to construct.
Using them for space travel would be tricky. Their extreme warping of spacetime leads to a wicked gravitational force that will be tough for spacetime travellers to handle. Also theory predicts that the tunnel would close off faster than even light could travel through it (turning both mouths in black holes). Some theorists have suggested that a special class of worm holes could be made to be traversable with the use of material that creates a negative pressure inside the Worm hole. They call this material exotic matter.
One reason that some scientists think that worm holes may not be allowed to exist by the Universe is that if they exist then time travel could be possible!
Yikes! What if you managed to go back in time and show up at the other mouth just before you entered it. Then you decide to stop yourself in the past from going in.
Uh Oh.... PARADOX!!!!!!!!!!!!!!!!
There are three ways out of the Paradox:
Want to read more? Check out a paper I wrote on the subject several years ago: Time Travel
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